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<p><dfn class="terminology">Note</dfn>(i) For a regular singular point, the corresponding power series solutions are called Frobenius solutions.(ii) The procedures to obtain series solution:(1) Identify the point concerned is an ordinary point or regular singular point.(2) Ordinary point: The independent solutions are given by</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=\sum_{n=0}^{\infty} a_n (x-x_0)^n=a_0 y_1+a_1 y_2 .
\end{equation*}
</div>
<p class="continuation">For <span class="process-math">\(y_1\text{,}\)</span> setting <span class="process-math">\(a_0=1, a_1=0\text{.}\)</span>For <span class="process-math">\(y_2\text{,}\)</span> setting <span class="process-math">\(a_0=0, a_1=1\text{.}\)</span>(3) Regular singular point at <span class="process-math">\(x_0\text{:}\)</span>A change of variables: <span class="process-math">\(t=x-x_0\text{,}\)</span> which changes the regular singular point to <span class="process-math">\(t=0\text{.}\)</span>(3 a) Find <span class="process-math">\(p_0\)</span> and <span class="process-math">\(q_0\)</span> and solve the indicial equations:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
r^2 +r (p_0-1)+q_0=0
\end{equation*}
</div>
<p class="continuation">to obtain two roots <span class="process-math">\(r_1\)</span> and <span class="process-math">\(r_2\)</span> (set <span class="process-math">\(r_1 \geq r_2\)</span>).(3 b) Construct the first solution as</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
y=t^{r_1} \sum_{n=0}^{\infty} a_n t^n
\end{equation*}
</div>
<p class="continuation">where <span class="process-math">\(a_0=1\text{.}\)</span>(3 c) To construct the second solution: there are three cases.</p>
<span class="incontext"><a href="sec5_5.html#p-242" class="internal">in-context</a></span>
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